Inverse problems for a generalized subdiffusion equation with final overdetermination
Abstract
We consider two inverse problems for a generalized subdiffusion equation that use the final overdetermination condition. Firstly, we study a problem of reconstruction of a specific space-dependent component in a source term. We prove existence, uniqueness and stability of the solution to this problem. Based on these results, we consider an inverse problem of identification of a space-dependent coefficient of a linear reaction term. We prove the uniqueness and local existence and stability of the solution to this problem.
Keyword : inverse problem, subdiffusion, final overdetermination, fractional diffusion
How to Cite
Kinash, N., & Janno, J. (2019). Inverse problems for a generalized subdiffusion equation with final overdetermination. Mathematical Modelling and Analysis, 24(2), 236-262. https://doi.org/10.3846/mma.2019.016
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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J. Janno and K. Kasemets. A positivity principle for parabolic integro-differentialequations and inverse problems with final overdetermination. Inverse Problems and Imaging, 3(1):17–41, 2009. https://doi.org/10.3934/ipi.2009.3.17
J. Janno and K. Kasemets. Identification of a kernel in an evolutionary integralequation occurring in subdiffusion. Journal of Inverse and Ill-posed Problems, 25(6):777–798, 2017. https://doi.org/10.1515/jiip-2016-0082
J. Janno and K. Kasemets. Uniqueness for an inverse problem for a semilineartime-fractional diffusion equation. Inverse Problems and Imaging, 11(1):125–149, 2017. https://doi.org/10.3934/ipi.2017007
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F. Mainardi, A. Mura, G. Pagnini and R. Gorenflo. Time-fractional diffusion ofdistributed order. Journal of Vibration and Control, 14(9-10):1267–1290, 2008. https://doi.org/10.1177/1077546307087452
D.G. Orlovsky. Parameter determination in a differential equation of fractionalorder with Riemann-Liouville fractional derivative in a Hilbert space. Journal of Siberian Federal University, 8(1):55–63, 2015. https://doi.org/10.17516/1997-1397-2015-8-1-55-63
Y.Povstenko. Fractional heat conduction and associated thermal stress. Journal of Thermal Stresses, 28(1):83–102,2004. https://doi.org/10.1080/014957390523741
J. Prüss. Evolutionary Integral Equations and Applications. Birkhäuser Verlag, Basel, 1993. https://doi.org/10.1007/978-3-0348-8570-6
F. Sabzikar, M. M. Meerschaert and J.Chen. Tempered fractional calculus. Journal of Computational Physics, 293:14–28, 2015. https://doi.org/10.1016/j.jcp.2014.04.024
K. Sakamoto and M. Yamamoto. Inverse source problem with a final overde-termination for a fractional diffusion equation. Mathematical Control & Related Fields, 1(4):509–518, 2011. https://doi.org/10.3934/mcrf.2011.1.509
I.M. Sokolov, A.V. Chechkin and J. Klafter. Distributed-order fractional kinetics. Acta Physica Polonica B, 35(47):1323–1341, 2004.
S. Tatar and S. Ulusoy. An inverse source problem for a one-dimensional space-time fractional diffusion equation. Applicable Analysis, 94(11):2233–2244, 2015. https://doi.org/10.1080/00036811.2014.979808
V. Vergara and R. Zacher. Stability, instability, and blowup for time fractionaland other nonlocal in time semilinear subdiffusion equations. Journal of Evolution Equations, 17(1):599–626, 2017. https://doi.org/10.1007/s00028-016-0370-2
R. Zacher. Weak solutions of abstract evolutionary integro-differential equations in Hilbert spaces. Funkcialaj Ekvacioj, 52(1):1–18, 2009. https://doi.org/10.1619/fesi.52.1
E. Beretta and C. Cavaterra. Identifying a space dependent coefficient in areaction-diffusion equation. Inverse Problems and Imaging, 5(2):285–296, 2011. https://doi.org/10.3934/ipi.2011.5.285
A.V. Chechkin, V.Yu. Gonchar, R. Gorenflo, N. Korabel and I.M.Sokolov. Generalized fractional diffusion equations for accelerating subd-iffusion and truncated Lévy flights. Physical Review E, 78:021111, 2008. https://doi.org/10.1103/PhysRevE.78.021111
A.V. Chechkin, R. Gorenflo and I.M. Sokolov. Fractional diffusion ininhomogeneous media. J. Phys. A: Math. Gen., 38(42):679–684, 2005. https://doi.org/10.1088/0305-4470/38/42/L03
P.Clement and J.Prüss. Completely positive measures and Fellersemigroups. MathematischeAnnalen, 287(1):73–105,1990. https://doi.org/10.1007/BF01446879
D. Frömberg. Reaction Kinetics under Anomalous Diffusion. PhD thesis, Humboldt-Universität zu Berlin, Berlin, 8 1981.
K.M. Furati, O.S. Iyiola and M. Kirane. An inverse problem for a generalizedfractional diffusion. Applied Mathematics and Computation, 249:24–31, 2014. https://doi.org/10.1016/j.amc.2014.10.046
D. Gilbarg and N. Trudinger. Elliptic Partial Differential Equations of SecondOrder. Springer, Berlin, 1977. https://doi.org/10.1007/978-3-642-96379-7
G. Gripenberg. On Volterra equations of the first kind. Integral Equations and Operator Theory, 3(4):473–488, 1980. https://doi.org/10.1007/BF01702311
G. Gripenberg, S.-O. Londen and O. Staffans. Volterra Integral and Functional Equations. Cambridge University Press, Cambridge, 1990. https://doi.org/10.1017/CBO9780511662805
G.H. Hardy and J.E. Littlewood. Some properties of fractional integrals I. Math-ematische Zeitschrift, 27:565–606, 1928. https://doi.org/10.1007/BF01171116
V. Isakov. Inverse Problems for Partial Differential Equations. Springer, NewYork, 2006.
J. Janno and K. Kasemets. A positivity principle for parabolic integro-differentialequations and inverse problems with final overdetermination. Inverse Problems and Imaging, 3(1):17–41, 2009. https://doi.org/10.3934/ipi.2009.3.17
J. Janno and K. Kasemets. Identification of a kernel in an evolutionary integralequation occurring in subdiffusion. Journal of Inverse and Ill-posed Problems, 25(6):777–798, 2017. https://doi.org/10.1515/jiip-2016-0082
J. Janno and K. Kasemets. Uniqueness for an inverse problem for a semilineartime-fractional diffusion equation. Inverse Problems and Imaging, 11(1):125–149, 2017. https://doi.org/10.3934/ipi.2017007
K. Kasemets and J. Janno. Reconstruction of a source term in a parabolicintegro-differential equation from final data. Mathematical Modelling and Analysis, 16(2):199–219, 2011. https://doi.org/10.3846/13926292.2011.578282
N. Kinash and J. Janno. Inverse problems for a perturbed time fractional diffusion equation with final overdetermination. Math. Meth. Appl. Sci., 41(5):1925–1943, 2018. https://doi.org/10.1002/mma.4719
M. Kirane, S.A. Malik and M.A. Al-Gwaiz. An inverse source problem for a two dimensional time fractional diffusion equation with nonlocal boundary conditions. Math. Meth. Appl. Sci., 36(9):1056–1069, 2013. https://doi.org/10.1002/mma.2661
A.N. Kochubei. General fractional calculus, evolution equations, and renewal processes. Integral Equations and Operator Theory, 71(4):583–600, 2011. https://doi.org/10.1007/s00020-011-1918-8
Y. Liu, M. Yamamoto and W. Rundell. Strong maximum principle for fractional diffusion equations and an application to an inverse source problem. Fractional Calculus and Applied Analysis, 19(4), 2016. https://doi.org/10.1515/fca-2016-0048
A. Lopushansy and H. Lopushanska. Inverse source Cauchy problem for a timefractional diffusion-wave equation with distributions. Electronic Journal of Differential Equations, 2017(182):1–14, 2017.
Y. Luchko. Maximum principle and its application for the time-fractional diffusion equations. Fractional Calculus and Applied Analysis, 14(1):110–124, 2011. https://doi.org/10.2478/s13540-011-0008-6
Y. Luchko and M. Yamamoto. General time-fractional diffusion equation: Some uniqueness and existence results for the initial-boundary-value problems. Fractional Calculus and Applied Analysis, 19(3):676–695, 2016. https://doi.org/10.1515/fca-2016-0036
A. Lunardi. Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkhäuser, Basel, 1995.
F. Mainardi, A. Mura, G. Pagnini and R. Gorenflo. Time-fractional diffusion ofdistributed order. Journal of Vibration and Control, 14(9-10):1267–1290, 2008. https://doi.org/10.1177/1077546307087452
D.G. Orlovsky. Parameter determination in a differential equation of fractionalorder with Riemann-Liouville fractional derivative in a Hilbert space. Journal of Siberian Federal University, 8(1):55–63, 2015. https://doi.org/10.17516/1997-1397-2015-8-1-55-63
Y.Povstenko. Fractional heat conduction and associated thermal stress. Journal of Thermal Stresses, 28(1):83–102,2004. https://doi.org/10.1080/014957390523741
J. Prüss. Evolutionary Integral Equations and Applications. Birkhäuser Verlag, Basel, 1993. https://doi.org/10.1007/978-3-0348-8570-6
F. Sabzikar, M. M. Meerschaert and J.Chen. Tempered fractional calculus. Journal of Computational Physics, 293:14–28, 2015. https://doi.org/10.1016/j.jcp.2014.04.024
K. Sakamoto and M. Yamamoto. Inverse source problem with a final overde-termination for a fractional diffusion equation. Mathematical Control & Related Fields, 1(4):509–518, 2011. https://doi.org/10.3934/mcrf.2011.1.509
I.M. Sokolov, A.V. Chechkin and J. Klafter. Distributed-order fractional kinetics. Acta Physica Polonica B, 35(47):1323–1341, 2004.
S. Tatar and S. Ulusoy. An inverse source problem for a one-dimensional space-time fractional diffusion equation. Applicable Analysis, 94(11):2233–2244, 2015. https://doi.org/10.1080/00036811.2014.979808
V. Vergara and R. Zacher. Stability, instability, and blowup for time fractionaland other nonlocal in time semilinear subdiffusion equations. Journal of Evolution Equations, 17(1):599–626, 2017. https://doi.org/10.1007/s00028-016-0370-2
R. Zacher. Weak solutions of abstract evolutionary integro-differential equations in Hilbert spaces. Funkcialaj Ekvacioj, 52(1):1–18, 2009. https://doi.org/10.1619/fesi.52.1